Question

Six baseball throws are shown below. In each case the baseball is thrown at the same initial speed and from the same height above the ground. Assume that the effects of air resistance are negligible. Rank these throws according to the speed of the baseball the instant before it hits the ground.

Rank from largest to smallest. To rank items as equivalent, overlap them.

Answer 1

Concepts and reason

The concept required to solve this problem is conservation of energy.

Initially, calculate the kinetic and potential energy of the balls. Finally, rank the baseballs based on their respective final velocities.

Fundamentals

The law of conservation of energy states that the gain in the kinetic energy is equal to the loss in the potential energy.

The expression is as follows:

${U_i} - {U_f} = {K_f} - {K_i}$

Here, ${U_i}{\rm{ and }}{U_f}$ is the initial and final potential energy respectively and ${K_i}{\rm{ and }}{K_f}$ is the initial and final kinetic energy respectively.

The expression foe the kinetic energy is as follows:

$T = \frac{1}{2}m{v^2}$

Here, m is the mass of the object and v is the velocity of an object.

The expression for the potential energy is as follows:

$U = mgh$

Here, g is the acceleration due to gravity and h is the height.

The longer the ball is in the air, the longer it will take to hit the ground. And, larger the final velocity will be when the ball hits the ground.

The balls are thrown from the same height. Thus, the potential energy will be the same for all the balls. Also, when the ball reaches the ground the change in potential energy would also be the same.

According, to the conservation of energy the loss in potential energy is equal to the gain in kinetic energy. Hence the change in potential energy for the balls would be equal to the change in kinetic energy. Also, since the change in potential energy is the same for all the balls the change in kinetic energy will also be the same.

Further since the initial speed of all the balls are same, the final speed will also be the same.

Hence the speed of all the baseballs as they hit the ground would be the same.

Let all the baseballs be thrown from a height $h$ with an initial velocity $u$ .

For case1,

The loss in potential energy is,

$\Delta {U_1} = mgh$

The gain in kinetic energy is,

$\Delta {K_1} = \frac{1}{2}{m_1}{v_1}^2 - \frac{1}{2}{m_1}{u^2}$

Apply the law of conservation of energy.

$\Delta {U_1} = \Delta {K_1}$

Substitute $mgh$ for $\Delta {U_1}$ and $\frac{1}{2}{m_1}{v_1}^2 - \frac{1}{2}{m_1}{u^2}$

$\begin{array}{c}\\{m_1}gh = \frac{1}{2}{m_1}{v_1}^2 - \frac{1}{2}{m_1}{u^2}\\\\{v_1} = \sqrt {{u^2} + 2gh} \\\end{array}$

Here, ${v_1}$ is the final velocity and ${m_1}$ is the mass of the baseball.

Similarly, for case2,

The final velocity of baseball 2 will be,

${v_2} = \sqrt {{u^2} + 2gh}$

The final velocity of baseball 3 will be,

${v_3} = \sqrt {{u^2} + 2gh}$

The final velocity of baseball 4 will be,

${v_4} = \sqrt {{u^2} + 2gh}$

The final velocity of baseball 5 will be,

${v_5} = \sqrt {{u^2} + 2gh}$

The final velocity of baseball 6 will be,

${v_6} = \sqrt {{u^2} + 2gh}$

From the expression for ${v_1},{v_2},{v_3},{v_4},{v_5}{\rm{ and }}{v_6}$ it is clear that ${v_1} = {v_2} = {v_3} = {v_4} = {v_5} = {v_6}$ .

Ans:

The ranking of the six baseballs is ${v_1} = {v_2} = {v_3} = {v_4} = {v_5} = {v_6}$ .